= Quadratic longitudinal displacement of an inextensible planar filament
{title2=$X_{2,s}=-\tfrac12y_s^2$}
Write a small planar displacement in arclength coordinates as $X=s+X_2$ and $Y=y$. The <inextensibility> relation $X_s^2+Y_s^2=1$ gives $X_{2,s}=-y_s^2/2$ to quadratic order. Thus longitudinal material velocity enters one order later than the transverse velocity. For a periodic material stroke with no mean axial drift, its mean velocity vanishes, although its instantaneous contribution is needed for exact local force balance.
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